happyjack27 wrote:also i should add that cusps by definition (i believe) lie on the boundary of a volume. and the boundary of a volume is a surface thus they lie on a surface.
I am not sure about this, but it seems quite possible that hte definition of "cusp" changes with the basic subject, so a mathematical cusp may be different than a magnetic cusp, may be different than a gravitational cusp.
hmm, yes, good point. i looked it up and it appears that it does in most uses in any case refer to a boundary, and thus they in any case lie on a a manifold at least one dimension lower than that which they bounds (otherwise the manifold wouldn't bound). so the logic still holds even in the broader definitions. for instance, the "cusps" of a crescent moon are the points where the two curves outlining it ("bounding" it) meet.